An analytic solution for a fourier transform

1. The problem statement, all variables and given/known data
the function is Exp[-w^2]/w^2, how to solve the Fourier transform analytically with Residue theorem?
It is better if there is more general results.
Mathematica can solve it analytically, but I need a human-soluable way.

2. Relevant equations

3. The attempt at a solution
My attempt follows the Fresnel integral, but there is still something wrong.